Benjamin McKay

Soliton solutions for the Laplacian co-flow of some G2-structures with symmetry

Abstract We consider the Laplacian “co-flow” of G 2 -structures: ∂ ∂ t ψ = − Δ d ψ where ψ is the dual 4-form of a G 2 -structure φ and Δ d is the Hodge Laplacian on forms. Assuming short-time existence and uniqueness, this flow preserves the condition of the G 2 -structure being coclosed ( d ψ = 0 ). We study this flow for two explicit examples of coclosed G 2 -structures with symmetry. These are given by warped products of an interval or a circle with a compact 6-manifold N which is taken to be either a nearly Kahler manifold or a Calabi–Yau manifold. In both cases, we derive the flow equations and also the equations for soliton solutions. In the Calabi–Yau case, we find all the soliton solutions explicitly. In the nearly Kahler case, we find several special soliton solutions, and reduce the general problem to a single third order highly nonlinear ordinary differential equation.

Resting muscle sounds in anesthetized patients

It is known that contracting muscle makes low frequency sound vibrations. Small vibrations of uncertain origin are found over resting muscle. These could be shown to be of muscle origin if they significantly diminish in response to agents expected to decrease muscle activity. Thiopental, propofol, and neuromuscular-junction blocking muscle relaxants have such properties. Twenty-one subjects slated for elective surgery for which they would routinely be anesthetized and paralysed gave informed consent to having a small accelerometer taped upon their supine biceps (9 subjects), or volar forearm (12 subjects). Recordings were made in four stages while subjects: (i) lifted a 2-kg weight just off the sponge armrest on which their outstretched arm lay; (ii) relaxed their arm in the awake state prior to anesthesia; (iii) had anesthesia induced with intravenous thiopental (n = 11) or propofol (n = 10); and (iv) were paralysed. Recordings were digitised at 172-Hz and 6-s segments fast Fourier transformed (FFT). Total signal power, as determined by the area under the power spectrum, was significantly different (p < 0.05) in all stages for the biceps and in all but stages (iii) from (iv) in the forearm. It appears that resting muscle generates measurable vibrations.

Holomorphic Parabolic Geometries and Calabi{Yau Manifolds

We prove that the only complex parabolic geometries on Calabi{Yau manifolds are the homogeneous geometries on complex tori. We also classify the complex parabolic geometries on homogeneous compact Kahler manifolds.

Holomorphic Cartan geometries and rational curves

Abstract We prove that any compact Kähler manifold bearing a holomorphic Cartan geometry contains a rational curve just when the Cartan geometry is inherited from a holomorphic Cartan geometry on a lower dimensional compact Kähler manifold. This shows that many complex manifolds admit no or few holomorphic Cartan geometries.

Holomorphic Cartan geometries and Calabi-Yau manifolds

Let MM be a connected complex projective manifold such that c1(T(1,0)M)=0c1(T(1,0)M)=0. If MM admits a holomorphic Cartan geometry, then we show that MM is holomorphically covered by an abelian variety.

Extension Phenomena for Holomorphic Geometric Structures

The most commonly encountered types of complex analytic G-structures and Cartan geometries cannot have singularities of complex codimension 2 or more.

Sussmann's orbit theorem and maps☆

Abstract A map between manifolds which matches up families of complete vector fields is a fiber bundle mapping on each orbit of those vector fields.

Lagrangian submanifolds in affine symplectic geometry

Abstract We uncover the lowest order differential invariants of Lagrangian submanifolds under affine symplectic maps, and find out what happens when they are constant.

Almost complex rigidity of the complex projective plane

An isomorphism of symplectically tame smooth pseudocomplex structures on the complex projective plane which is a homeomorphism and differentiable of full rank at two points is smooth.

Symmetries of holomorphic geometric structures on tori

Abstract We prove that any holomorphic locally homogeneous geometric structure on a complex torus of dimension two, modelled on a complex homogeneous surface, is translation invariant. We conjecture that this result is true in any dimension. In higher dimension, we prove it for G nilpotent. We also prove that for any given complex algebraic homogeneous space (X, G), the translation invariant (X, G)-structures on tori form a union of connected components in the deformation space of (X, G)-structures.